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for (let i = 0; i < prefix.length; i++) { if (str[i] !== prefix[i]) return false; } return true; // Alternatively: return str.startsWith(prefix); } // Check if 'suffix' is a suffix of 'str' function isSuffix(str: string, suffix: string): boolean { if (suffix.length > str.length) return false; const offset = str.length - suffix.length; for (let i = 0; i < suffix.length; i++) { if (str[offset + i] !== suffix[i]) return false; } return true; // Alternatively: return str.endsWith(suffix); } // Check if 'sub' is a substring of 'str' function isSubstring(str: string, sub: string): boolean { if (sub.length === 0) return true; // Empty is always substring if (sub.length > str.length) return false; // Check all possible starting positions for (let i = 0; i <= str.length - sub.length; i++) { let match = true; for (let j = 0; j < sub.length; j++) { if (str[i + j] !== sub[j]) { match = false; break; } } if (match) return true; } return false; // Alternatively: return str.includes(sub); } // Generate all prefixes of a string function allPrefixes(str: string): string[] { const prefixes: string[] = ['']; // Empty prefix for (let i = 1; i <= str.length; i++) { prefixes.push(str.slice(0, i)); } return prefixes; } // Generate all suffixes of a string function allSuffixes(str: string): string[] { const suffixes: string[] = []; for (let i = 0; i <= str.length; i++) { suffixes.push(str.slice(i)); } return suffixes; // Includes empty suffix at end } // Example usage const S = "algorithm"; console.log("Prefixes:", allPrefixes(S)); console.log("Suffixes:", allSuffixes(S)); console.log("Is 'algo' a prefix?", isPrefix(S, "algo")); // true console.log("Is 'rithm' a suffix?", isSuffix(S, "rithm")); // true console.log("Is 'gori' a substring?", isSubstring(S, "gori")); // true73:T75f,""" Comprehensive prefix, suffix, and substring utilities These operations are fundamental to pattern matching algorithms """ from typing import List def is_prefix(s: str, prefix: str) -> bool: """Check if 'prefix' is a prefix of 's'""" if len(prefix) > len(s): return False for i in range(len(prefix)): if s[i] != prefix[i]: return False return True # Alternatively: return s.startswith(prefix) def is_suffix(s: str, suffix: str) -> bool: """Check if 'suffix' is a suffix of 's'""" if len(suffix) > len(s): return False offset = len(s) - len(suffix) for i in range(len(suffix)): if s[offset + i] != suffix[i]: return False return True # Alternatively: return s.endswith(suffix) def is_substring(s: str, sub: str) -> bool: """Check if 'sub' is a substring of 's'""" if len(sub) == 0: return True # Empty is always substring if len(sub) > len(s): return False # Check all possible starting positions for i in range(len(s) - len(sub) + 1): match = True for j in range(len(sub)): if s[i + j] != sub[j]: match = False break if match: return True return False # Alternatively: return sub in s def all_prefixes(s: str) -> List[str]: """Generate all prefixes of a string""" return [''] + [s[:i] for i in range(1, len(s) + 1)] def all_suffixes(s: str) -> List[str]: """Generate all suffixes of a string""" return [s[i:] for i in range(len(s) + 1)] # Example usage S = "algorithm" print("Prefixes:", all_prefixes(S)) print("Suffixes:", all_suffixes(S)) print("Is 'algo' a prefix?", is_prefix(S, "algo")) # True print("Is 'rithm' a suffix?", is_suffix(S, "rithm")) # True print("Is 'gori' a substring?", is_substring(S, "gori")) # True74:T846,/** * Formal implementation of the pattern matching problem * Returns all positions where pattern P occurs in text T */ interface PatternMatchResult { positions: number[]; // All valid shift positions comparisons: number; // Total character comparisons (for analysis) } /** * Naive pattern matching: directly implements the formal definition * Time Complexity: O(n × m) where n = |T| and m = |P| * Space Complexity: O(k) where k = number of matches */ function findPatternOccurrences(T: string, P: string): PatternMatchResult { const n = T.length; const m = P.length; const positions: number[] = []; let comparisons = 0; // Edge cases if (m === 0) { // Empty pattern matches at every position (including after last char) return { positions: Array.from({ length: n + 1 }, (_, i) => i), comparisons: 0 }; } if (m > n) { // Pattern longer than text: no matches possible return { positions: [], comparisons: 0 }; } // Check each valid starting position // Position i is valid if 0 ≤ i ≤ n - m (enough room for pattern) for (let i = 0; i <= n - m; i++) { // Check if P occurs at position i let isMatch = true; for (let j = 0; j < m; j++) { comparisons++; if (T[i + j] !== P[j]) { isMatch = false; break; } } if (isMatch) { positions.push(i); } } return { positions, comparisons }; } // Example usage const T = "abracadabra"; const P = "abra"; const result = findPatternOccurrences(T, P); console.log(`Text: "${T}" (length ${T.length})`); console.log(`Pattern: "${P}" (length ${P.length})`); console.log(`Occurrences at positions: [${result.positions.join(', ')}]`); console.log(`Total comparisons: ${result.comparisons}`); // Verify each match result.positions.forEach(pos => { const match = T.slice(pos, pos + P.length); console.log(` Position ${pos}: "${match}" === "${P}" ? ${match === P}`); });75:T7ae,""" Formal implementation of the pattern matching problem Returns all positions where pattern P occurs in text T """ from typing import List, NamedTuple class PatternMatchResult(NamedTuple): positions: List[int] # All valid shift positions comparisons: int # Total character comparisons (for analysis) def find_pattern_occurrences(T: str, P: str) -> PatternMatchResult: """ Naive pattern matching: directly implements the formal definition Time Complexity: O(n × m) where n = |T| and m = |P| Space Complexity: O(k) where k = number of matches """ n = len(T) m = len(P) positions: List[int] = [] comparisons = 0 # Edge cases if m == 0: # Empty pattern matches at every position (including after last char) return PatternMatchResult( positions=list(range(n + 1)), comparisons=0 ) if m > n: # Pattern longer than text: no matches possible return PatternMatchResult(positions=[], comparisons=0) # Check each valid starting position # Position i is valid if 0 ≤ i ≤ n - m (enough room for pattern) for i in range(n - m + 1): # Check if P occurs at position i is_match = True for j in range(m): comparisons += 1 if T[i + j] != P[j]: is_match = False break if is_match: positions.append(i) return PatternMatchResult(positions=positions, comparisons=comparisons) # Example usage T = "abracadabra" P = "abra" result = find_pattern_occurrences(T, P) print(f'Text: "{T}" (length {len(T)})') print(f'Pattern: "{P}" (length {len(P)})') print(f'Occurrences at positions: {result.positions}') print(f'Total comparisons: {result.comparisons}') # Verify each match for pos in result.positions: match = T[pos:pos + len(P)] print(f' Position {pos}: "{match}" === "{P}" ? {match == P}')76:T41c,The basic pattern matching problem has several variants depending on what output is required. Understanding these variants is crucial because different algorithms optimize for different outputs. **Variant 1: Decision Problem** *Question:* Does pattern P occur in text T at all? *Output:* Boolean (yes/no) *Complexity goal:* Often can short-circuit after finding first match, making average case faster. --- **Variant 2: First Occurrence** *Question:* What is the first (leftmost) position where P occurs in T? *Output:* Single position, or -1 if no occurrence *Complexity goal:* Can stop immediately upon finding first match. --- **Variant 3: Count Occurrences** *Question:* How many times does P occur in T? *Output:* Non-negative integer *Note:* Overlapping occurrences may or may not be counted depending on requirements. --- **Variant 4: All Occurrences** *Question:* List all positions where P occurs in T. *Output:* List of positions (may be empty) *Complexity goal:* Must examine entire text; output can be O(n) in worst case.77:T937,/** * All variants of the pattern matching problem */ // Variant 1: Decision - Does pattern occur? function containsPattern(T: string, P: string): boolean { const n = T.length; const m = P.length; if (m === 0) return true; if (m > n) return false; for (let i = 0; i <= n - m; i++) { let match = true; for (let j = 0; j < m && match; j++) { if (T[i + j] !== P[j]) match = false; } if (match) return true; // Early exit! } return false; } // Variant 2: First occurrence position (-1 if not found) function findFirstOccurrence(T: string, P: string): number { const n = T.length; const m = P.length; if (m === 0) return 0; // Empty pattern at start if (m > n) return -1; for (let i = 0; i <= n - m; i++) { let match = true; for (let j = 0; j < m && match; j++) { if (T[i + j] !== P[j]) match = false; } if (match) return i; // Early exit! } return -1; } // Variant 3: Count occurrences (including overlapping) function countOccurrences(T: string, P: string): number { const n = T.length; const m = P.length; if (m === 0) return n + 1; // Empty matches everywhere if (m > n) return 0; let count = 0; for (let i = 0; i <= n - m; i++) { let match = true; for (let j = 0; j < m && match; j++) { if (T[i + j] !== P[j]) match = false; } if (match) count++; } return count; } // Variant 3b: Count non-overlapping occurrences function countNonOverlapping(T: string, P: string): number { const n = T.length; const m = P.length; if (m === 0) return n + 1; if (m > n) return 0; let count = 0; let i = 0; while (i <= n - m) { let match = true; for (let j = 0; j < m && match; j++) { if (T[i + j] !== P[j]) match = false; } if (match) { count++; i += m; // Skip past this match } else { i++; } } return count; } // Example demonstrating overlapping const T = "aaaa"; const P = "aa"; console.log(`Text: "${T}", Pattern: "${P}"`); console.log(`Overlapping count: ${countOccurrences(T, P)}`); // 3 console.log(`Non-overlapping count: ${countNonOverlapping(T, P)}`); // 278:T855,""" All variants of the pattern matching problem """ def contains_pattern(T: str, P: str) -> bool: """Variant 1: Decision - Does pattern occur?""" n, m = len(T), len(P) if m == 0: return True if m > n: return False for i in range(n - m + 1): match = True for j in range(m): if T[i + j] != P[j]: match = False break if match: return True # Early exit! return False def find_first_occurrence(T: str, P: str) -> int: """Variant 2: First occurrence position (-1 if not found)""" n, m = len(T), len(P) if m == 0: return 0 # Empty pattern at start if m > n: return -1 for i in range(n - m + 1): match = True for j in range(m): if T[i + j] != P[j]: match = False break if match: return i # Early exit! return -1 def count_occurrences(T: str, P: str) -> int: """Variant 3: Count occurrences (including overlapping)""" n, m = len(T), len(P) if m == 0: return n + 1 # Empty matches everywhere if m > n: return 0 count = 0 for i in range(n - m + 1): match = True for j in range(m): if T[i + j] != P[j]: match = False break if match: count += 1 return count def count_non_overlapping(T: str, P: str) -> int: """Variant 3b: Count non-overlapping occurrences""" n, m = len(T), len(P) if m == 0: return n + 1 if m > n: return 0 count = 0 i = 0 while i <= n - m: match = True for j in range(m): if T[i + j] != P[j]: match = False break if match: count += 1 i += m # Skip past this match else: i += 1 return count # Example demonstrating overlapping T = "aaaa" P = "aa" print(f'Text: "{T}", Pattern: "{P}"') print(f'Overlapping count: {count_occurrences(T, P)}') # 3 print(f'Non-overlapping count: {count_non_overlapping(T, P)}') # 279:T485,The naive algorithm directly implements our formal definition, but its performance characteristics reveal why we need sophisticated pattern matching algorithms. **Time Complexity Analysis:** - **Outer loop:** Runs at most n - m + 1 times (each valid starting position) - **Inner loop:** Runs at most m times per outer iteration - **Worst case:** O((n - m + 1) × m) = O(n × m) For n = 1,000,000 and m = 1,000, that's up to **1 billion** comparisons. **Worst-case input:** ``` Text T: "aaaaaaaaaaaa...aaaab" (n-1 a's followed by b) Pattern P: "aaa...aaab" (m-1 a's followed by b) ``` For each of the n - m + 1 positions, the algorithm compares m - 1 characters (all matching 'a's) before the final mismatch at 'b'. This is O(n × m) in practice. **Average case:** For random text and pattern over a non-trivial alphabet, the expected comparisons per position is much smaller (mismatches happen early). For random strings, average case is O(n). But we can't rely on average case: - DNA sequences have only 4 characters (A, C, G, T) - Natural language has repetitive patterns - Adversarial inputs (attacks) can exploit worst-case behavior7a:T53c,Pattern matching can also be viewed through the lens of **shifts**. This perspective is particularly useful for understanding algorithms like Boyer-Moore. **Definition (Shift):** A **shift** s is the number of positions the pattern is shifted right relative to the text. Pattern P with shift s is aligned so that P[0] is compared with T[s]. **Valid shift:** A shift s is valid if P matches T starting at position s, i.e., T[s..s+m-1] = P. **The pattern matching problem restated:** Find all valid shifts s where 0 ≤ s ≤ n - m. **Visual representation of shifts:** ``` Text: a b r a c a d a b r a 0 1 2 3 4 5 6 7 8 9 10 Shift 0: a b r a ✓ Valid ↑ ↑ ↑ ↑ Shift 1: b r a c ✗ Invalid (P[0]='a' ≠ T[1]='b') ↑ Shift 7: a b r a ✓ Valid ↑ ↑ ↑ ↑ ``` **Why the shift perspective matters:** The naive algorithm increments shift by 1 after each comparison (regardless of what was learned). Advanced algorithms compute **larger shifts** when a mismatch occurs, skipping positions that cannot possibly be valid. - **KMP:** Uses the structure of the pattern to determine safe shifts - **Boyer-Moore:** Uses mismatch character and good suffix information - **Rabin-Karp:** Uses hashing to quickly skip non-matching positions7b:T63f,/** * Pattern matching from the shift perspective * This formulation directly expresses "find all valid shifts" */ function findValidShifts(T: string, P: string): number[] { const n = T.length; const m = P.length; const validShifts: number[] = []; // Range of possible shifts: 0 to n-m inclusive const maxShift = n - m; if (maxShift < 0) { return []; // Pattern longer than text } // Check each shift for (let s = 0; s <= maxShift; s++) { if (isValidShift(T, P, s)) { validShifts.push(s); } } return validShifts; } /** * Check if shift s is valid (pattern matches at position s) */ function isValidShift(T: string, P: string, s: number): boolean { const m = P.length; for (let j = 0; j < m; j++) { if (T[s + j] !== P[j]) { return false; } } return true; } /** * Visualize the matching process with shifts */ function visualizeShifts(T: string, P: string): void { const n = T.length; const m = P.length; const maxShift = n - m; console.log(`Text: ${T}`); console.log(` ${Array.from({length: n}, (_, i) => i % 10).join('')}`); console.log(''); for (let s = 0; s <= maxShift; s++) { const valid = isValidShift(T, P, s); const padding = ' '.repeat(s); const status = valid ? 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Must characters be consecutive?\n- Do overlapping occurrences count separately?\n- How do we describe positions within the text?\n- What if we're searching in binary data rather than English text?\n\nA formal definition answers all these questions unambiguously, providing precision that enables algorithm design and correctness proofs."},"type":"text"},{"data":{"icon":"Lightbulb","title":"Why Formalization Matters","content":"Informal descriptions lead to edge case bugs and implementation ambiguity. When Donald Knuth, James Morris, and Vaughan Pratt developed the KMP algorithm, their formal problem definition allowed them to prove their algorithm correct and analyze its complexity precisely. Without formalization, we cannot rigorously compare algorithms or guarantee correctness.","variant":"tip"},"type":"callout"},{"data":{"content":"**The journey from informal to formal:**\n\nWe will now build our formal definition piece by piece:\n\n1. **Alphabets** — The set of symbols we can use\n2. **Strings** — Finite sequences over an alphabet\n3. **Text and Pattern** — The specific strings in our problem\n4. **Position and Index** — How we reference locations\n5. **Occurrence/Match** — The precise condition for a match\n6. **The Pattern Matching Problem** — The complete formal statement\n\nEach definition builds on the previous, creating a rigorous foundation for all string algorithms."},"type":"text"}],"titleType":"number","titleNumber":1},{"id":"alphabets-and-strings","title":"Alphabets and Strings: The Building Blocks","content":[{"data":{"content":"Every string algorithm operates over an **alphabet**—the set of allowed symbols. This seemingly simple concept has profound implications for algorithm design and complexity analysis.\n\n**Definition (Alphabet):**\n\nAn **alphabet**, denoted **Σ** (sigma), is a finite, non-empty set of distinct symbols (also called characters or letters).\n\n**Examples of alphabets:**"},"type":"text"},{"data":{"rows":[["Binary","{0, 1}","2","",""],["DNA","{A, C, G, T}","4","",""],["Lowercase English","{a, b, c, ..., z}","26","",""],["ASCII","{all ASCII characters}","128","",""],["Unicode","{all Unicode code points}",">143,000","",""]],"headers":["Alphabet Name","Σ","Size","Σ",""]},"type":"table"},{"data":{"content":"The size of the alphabet, denoted **|Σ|**, directly impacts algorithm complexity for many string algorithms."},"type":"text"},{"data":{"icon":"Info","title":"Alphabet Size Matters","content":"Many string algorithms have complexity that depends on |Σ|. For DNA (|Σ| = 4), a brute-force approach might be acceptable. For Unicode (|Σ| > 143,000), the same approach becomes impractical. This is why algorithms like KMP, which are alphabet-independent, are so valuable.","variant":"note"},"type":"callout"},{"data":{"content":"**Definition (String):**\n\nA **string** over alphabet Σ is a finite sequence of symbols from Σ. The set of all strings over Σ is denoted **Σ*** (Σ-star).\n\n**Formal notation:**\n\nA string S of length n is written as:\n\n**S = S[0]S[1]S[2]...S[n-1]**\n\nwhere each S[i] ∈ Σ for 0 ≤ i < n.\n\n**Key string properties:**\n\n- **Length**: The number of symbols in S, denoted **|S|** or **len(S)**\n- **Empty string**: The unique string of length 0, denoted **ε** (epsilon) or **\"\"**\n- **Concatenation**: Joining strings S and T produces ST (or S·T)\n- **Substring**: Any contiguous subsequence of S\n- **Prefix**: A substring starting at position 0\n- **Suffix**: A substring ending at the last position"},"type":"text"},{"data":{"fileName":"string-definitions-example","languages":[{"code":"Alphabet: Σ = {a, b, c, ..., z}\n\nString: S = \"algorithm\"\nLength: |S| = 9\n\nIndexing (0-based):\n S[0] = 'a'\n S[1] = 'l'\n S[2] = 'g'\n ...\n S[8] = 'm'\n\nSubstrings of \"algorithm\":\n \"algo\" - prefix (positions 0-3)\n \"rithm\" - suffix (positions 4-8)\n \"gori\" - general substring (positions 2-5)\n \"algorithm\" - the string itself is both prefix and suffix\n \"\" - empty string (also valid prefix/suffix)\n\nConcatenation:\n \"algo\" · \"rithm\" = \"algorithm\"","label":"Conceptual","value":"plaintext"},{"code":"// String operations in TypeScript\nconst alphabet: Set = new Set('abcdefghijklmnopqrstuvwxyz'.split(''));\n\nconst S: string = \"algorithm\";\n\n// Length\nconsole.log(`|S| = ${S.length}`); // 9\n\n// Indexing (0-based)\nconsole.log(`S[0] = '${S[0]}'`); // 'a'\nconsole.log(`S[8] = '${S[8]}'`); // 'm'\n\n// Substrings using slice(start, end) - end is exclusive\nconst prefix: string = S.slice(0, 4); // \"algo\"\nconst suffix: string = S.slice(4); // \"rithm\"\nconst substring: string = S.slice(2, 6); // \"gori\"\n\n// Check if character is in alphabet\nconst isValidChar = (c: string): boolean => alphabet.has(c);\nconsole.log(isValidChar('a')); // true\nconsole.log(isValidChar('1')); // false\n\n// Concatenation\nconst concat: string = \"algo\" + \"rithm\"; // \"algorithm\"","label":"TypeScript","value":"typescript"},{"code":"# String operations in Python\nimport string\n\nalphabet = set(string.ascii_lowercase)\n\nS = \"algorithm\"\n\n# Length\nprint(f\"|S| = {len(S)}\") # 9\n\n# Indexing (0-based)\nprint(f\"S[0] = '{S[0]}'\") # 'a'\nprint(f\"S[8] = '{S[8]}'\") # 'm'\n\n# Substrings using slicing [start:end] - end is exclusive\nprefix = S[0:4] # \"algo\"\nsuffix = S[4:] # \"rithm\"\nsubstring = S[2:6] # \"gori\"\n\n# Check if character is in alphabet\ndef is_valid_char(c: str) -> bool:\n return c in alphabet\n\nprint(is_valid_char('a')) # True\nprint(is_valid_char('1')) # False\n\n# Concatenation\nconcat = \"algo\" + \"rithm\" # \"algorithm\"","label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"}],"titleType":"number","titleNumber":2},{"id":"substrings-prefixes-suffixes","title":"Substrings, Prefixes, and Suffixes","content":[{"data":{"content":"These three concepts form the vocabulary of pattern matching. Understanding them precisely is essential for grasping why certain algorithms work.\n\n**Definition (Substring):**\n\nString P is a **substring** of string S if there exist strings x and y such that:\n\n**S = xPy**\n\nwhere x and y can be empty. Equivalently, P = S[i..j] for some 0 ≤ i ≤ j ≤ |S|.\n\nThe substring starting at position i with length m is:\n\n**S[i..i+m-1] = S[i]S[i+1]...S[i+m-1]**\n\n**Definition (Prefix):**\n\nString P is a **prefix** of string S if there exists string y such that:\n\n**S = Py**\n\nEquivalently, P = S[0..k-1] for some 0 ≤ k ≤ |S|. A prefix is a substring that starts at position 0.\n\n**Definition (Suffix):**\n\nString P is a **suffix** of string S if there exists string x such that:\n\n**S = xP**\n\nEquivalently, P = S[k..|S|-1] for some 0 ≤ k ≤ |S|. A suffix is a substring that ends at the last position."},"type":"text"},{"data":{"rows":[["0","ε (empty)","ε (empty)"],["1","a","m"],["2","al","hm"],["3","alg","thm"],["4","algo","ithm"],["5","algor","rithm"],["6","algori","orithm"],["7","algorit","gorithm"],["8","algorith","lgorithm"],["9","algorithm","algorithm"]],"caption":"All Prefixes and Suffixes of 'algorithm'","headers":["Length","Prefix","Suffix"]},"type":"table"},{"data":{"icon":"Lightbulb","title":"Proper Substrings, Prefixes, and Suffixes","content":"A **proper** prefix (or suffix) of S is one that is not equal to S itself. Similarly, a **proper** substring excludes S as a whole. These distinctions become crucial when analyzing algorithms like KMP, where we specifically examine proper prefixes that are also suffixes.","variant":"tip"},"type":"callout"},{"data":{"content":"**Counting substrings:**\n\nFor a string of length n:\n\n- **Number of substrings** (including empty): 1 + 2 + ... + n + 1 = (n+1)(n+2)/2 = O(n²)\n- **Number of distinct substrings**: Between n+1 (e.g., \"aaa\") and n(n+1)/2 + 1 (all unique)\n- **Number of prefixes**: Exactly n + 1 (including empty)\n- **Number of suffixes**: Exactly n + 1 (including empty)\n\nThis quadratic growth in substrings explains why brute-force substring enumeration doesn't scale, and why data structures like suffix arrays (which store all n suffixes) are so powerful."},"type":"text"},{"data":{"fileName":"prefix-suffix-operations","languages":[{"code":"$72","label":"TypeScript","value":"typescript"},{"code":"$73","label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"}],"titleType":"number","titleNumber":3},{"id":"formal-problem-statement","title":"The Formal Pattern Matching Problem","content":[{"data":{"content":"With our vocabulary established, we can now state the pattern matching problem with mathematical precision.\n\n**Definition (Pattern Matching Problem):**\n\n**Given:**\n- A **text** T of length n over alphabet Σ: T = T[0]T[1]...T[n-1]\n- A **pattern** P of length m over the same alphabet Σ: P = P[0]P[1]...P[m-1]\n- Typically, m ≤ n (the pattern is not longer than the text)\n\n**Find:**\n- All positions i (where 0 ≤ i ≤ n - m) such that P **occurs** in T starting at position i\n\n**Definition (Occurrence/Match):**\n\nPattern P **occurs** (or **matches**) at position i in text T if and only if:\n\n**T[i + j] = P[j] for all j where 0 ≤ j < m**\n\nIn other words, the substring T[i..i+m-1] equals P.\n\nPosition i is called a **valid shift** or **match position**."},"type":"text"},{"data":{"icon":"Info","title":"Index Convention","content":"We use 0-based indexing throughout (positions start at 0). Some textbooks use 1-based indexing (positions start at 1). When reading algorithm papers, always verify which convention is used. The core algorithms are identical; only the index arithmetic differs slightly.","variant":"note"},"type":"callout"},{"data":{"content":"**Visual representation:**\n\nLet's visualize pattern matching with a concrete example:\n\n```\nText T: \"abracadabra\"\n 01234567890\n \nPattern P: \"abra\"\n 0123\n\nQuestion: At which positions does P occur in T?\n```\n\n**Checking each position:**"},"type":"text"},{"data":{"rows":[["0","\"abra\"","✓ Yes"],["1","\"brac\"","✗ No"],["2","\"raca\"","✗ No"],["3","\"acad\"","✗ No"],["4","\"cada\"","✗ No"],["5","\"adab\"","✗ No"],["6","\"dabr\"","✗ No"],["7","\"abra\"","✓ Yes"],["8+","(too short)","N/A"]],"headers":["Position i","Substring T[i..i+3]","Matches P?"]},"type":"table"},{"data":{"content":"**Answer:** P occurs at positions **0** and **7**."},"type":"text"},{"data":{"fileName":"formal-pattern-matching","languages":[{"code":"$74","label":"TypeScript","value":"typescript"},{"code":"$75","label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"}],"titleType":"number","titleNumber":4},{"id":"problem-output-variants","title":"Problem Variants and Output Specifications","content":[{"data":{"content":"$76"},"type":"text"},{"data":{"icon":"AlertCircle","title":"Overlapping Occurrences","content":"Overlapping occurrences occur when the pattern can match starting at positions that are closer together than the pattern length. For example, pattern 'aa' in text 'aaa' has overlapping occurrences at positions 0 and 1. Most formal definitions count overlapping occurrences separately, but some applications (like word counting in natural language) may want non-overlapping matches only.","variant":"warning"},"type":"callout"},{"data":{"fileName":"pattern-matching-variants","languages":[{"code":"$77","label":"TypeScript","value":"typescript"},{"code":"$78","label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"}],"titleType":"number","titleNumber":5},{"id":"complexity-considerations","title":"Complexity Analysis: Why Naive Isn't Enough","content":[{"data":{"content":"$79"},"type":"text"},{"data":{"rows":[["1,000","10","~10,000","~0.01 ms"],["10,000","100","~1,000,000","~1 ms"],["100,000","1,000","~100,000,000","~100 ms"],["1,000,000","10,000","~10,000,000,000","~10 sec"],["10,000,000","100,000","~1,000,000,000,000","~17 min"]],"caption":"Naive Algorithm: Concrete Performance Examples","headers":["Text Length (n)","Pattern Length (m)","Worst-Case Comparisons","Time at 10⁹ ops/sec"]},"type":"table"},{"data":{"icon":"AlertCircle","title":"The Scale Problem","content":"A search engine processes billions of queries daily. If each search in a 1MB document took 10 seconds, the system would be unusable. This is why O(n + m) algorithms like KMP, Rabin-Karp, and Boyer-Moore are essential—they provide the massive speedup required for real-world applications.","variant":"error"},"type":"callout"},{"data":{"content":"**Space Complexity:**\n\n- **Naive algorithm:** O(1) extra space (just loop variables and output)\n- **Advanced algorithms:** Trade space for time\n - KMP: O(m) for failure function\n - Rabin-Karp: O(1) for hash computation\n - Suffix array: O(n) for preprocessed data structure\n - Suffix tree: O(n) but with larger constants\n\n**The fundamental insight:**\n\nThe naive algorithm wastes work by forgetting what it learned. When a mismatch occurs at position j, we discard all information about the j characters we already matched and start fresh. Efficient algorithms exploit the structure of the pattern to avoid redundant comparisons.\n\nThis insight—that we can **preprocess the pattern** to avoid redundant text examination—is the key to all efficient pattern matching algorithms."},"type":"text"}],"titleType":"number","titleNumber":6},{"id":"shift-perspective","title":"The Shift Perspective: Another Way to View Pattern Matching","content":[{"data":{"content":"$7a"},"type":"text"},{"data":{"fileName":"shift-based-matching","languages":[{"code":"$7b","label":"TypeScript","value":"typescript"}],"showLineNumbers":true},"type":"code"}],"titleType":"number","titleNumber":7},{"id":"summary-insights","title":"Summary: The Formal Foundation","content":[{"data":{"content":"We have established the precise mathematical foundation for pattern matching. This formalization is not mere academic exercise—it enables rigorous algorithm design, complexity analysis, and correctness proofs."},"type":"text"},{"data":{"type":"ordered","items":["**Alphabet (Σ):** A finite set of symbols; its size affects algorithm complexity","**String:** A finite sequence of symbols from an alphabet","**Substring, Prefix, Suffix:** Contiguous parts of a string with specific positions","**Text (T) and Pattern (P):** The specific strings in our problem; |T| = n, |P| = m","**Occurrence/Match:** Pattern P occurs at position i if T[i..i+m-1] = P","**Valid Shift:** Position s where pattern matches text"],"title":{"icon":"CheckCircle","text":"Key Definitions Established","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"items":["**The naive algorithm is O(n × m)** — acceptable for small inputs but catastrophic at scale","**Overlapping occurrences matter** — the same text position can participate in multiple matches","**Problem variants affect optimization** — first-match can exit early; all-matches cannot","**The shift perspective** reveals optimization opportunities that lead to efficient algorithms","**Preprocessing is key** — efficient algorithms preprocess the pattern to enable larger shifts"],"title":{"icon":"Target","text":"Critical Insights","type":"icon","color":"primary","variant":"outline-circle"},"variant":"glass"},"type":"list"},{"data":{"icon":"CheckCircle","title":"Foundation Complete","content":"You now possess the formal vocabulary and definitions required to understand all pattern matching algorithms. The concepts of text, pattern, occurrence, and shift will recur throughout this chapter. Next, we'll examine the specific components—text, pattern, and occurrence—in greater depth to understand the nuances that efficient algorithms exploit.","variant":"success"},"type":"callout"},{"data":{"content":"**Looking ahead:**\n\nWith our formal foundation established, the next pages will explore:\n\n- **Text, Pattern, and Occurrence (Page 2):** Deep dive into the relationship between these elements\n- **Single vs Multiple Pattern Matching (Page 3):** When we search for one pattern vs. many patterns simultaneously\n- **Applications (Page 4):** Real-world systems built on pattern matching—from search engines to genome analysis\n\nEach page builds on this formal groundwork, showing how precise definitions enable elegant and efficient solutions."},"type":"text"}],"titleType":"number","titleNumber":8}],"isLocked":false},{"id":"seed_135e4efc68a7977916227954e6e5e081","title":"Text, Pattern, and Occurrence — The Core Elements","order":1,"content":null,"isLocked":true},{"id":"seed_5373b438191f947e0ea38c8e56006c30","title":"Single vs Multiple Pattern Matching","order":2,"content":null,"isLocked":true},{"id":"seed_1b571028d57dcdca8fcdc4bceb7c8d40","title":"Applications: Search Engines, Bioinformatics, Text Editors","order":3,"content":null,"isLocked":true}]}},"dataUpdateCount":1,"dataUpdatedAt":1792280222013,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getModule"],{"input":{"slug":"string-matching-problem-formalization"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getModule\"],{\"input\":{\"slug\":\"string-matching-problem-formalization\"},\"type\":\"query\"}]"},{"dehydratedAt":1792280222055,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792280222013,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getProgress"],{"input":{"moduleId":"seed_af9b72bafb17a8badb2ac2aa4776d6cf"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getProgress\"],{\"input\":{\"moduleId\":\"seed_af9b72bafb17a8badb2ac2aa4776d6cf\"},\"type\":\"query\"}]"},{"dehydratedAt":1792280222055,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792280222013,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["feedback","get"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_135e4efc68a7977916227954e6e5e081"},"type":"query"}],"queryHash":"[[\"feedback\",\"get\"],{\"input\":{\"contentId\":\"seed_135e4efc68a7977916227954e6e5e081\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792280222055,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792280222013,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["feedback","getUserLike"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_135e4efc68a7977916227954e6e5e081"},"type":"query"}],"queryHash":"[[\"feedback\",\"getUserLike\"],{\"input\":{\"contentId\":\"seed_135e4efc68a7977916227954e6e5e081\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792280222055,"state":{"data":{"json":[]},"dataUpdateCount":1,"dataUpdatedAt":1792280222013,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["highlights","getHighlights"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_135e4efc68a7977916227954e6e5e081"},"type":"query"}],"queryHash":"[[\"highlights\",\"getHighlights\"],{\"input\":{\"contentId\":\"seed_135e4efc68a7977916227954e6e5e081\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792280222055,"state":{"data":"$undefined","dataUpdateCount":0,"dataUpdatedAt":0,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"pending","fetchStatus":"fetching"},"queryKey":[["highlights","getPopularHighlights"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_135e4efc68a7977916227954e6e5e081","minCount":2},"type":"query"}],"queryHash":"[[\"highlights\",\"getPopularHighlights\"],{\"input\":{\"contentId\":\"seed_135e4efc68a7977916227954e6e5e081\",\"contentType\":\"LEARNING_PAGE\",\"minCount\":2},\"type\":\"query\"}]","promise":"$@7c"}]},"children":"$L7d"}]